November  2013, 12(6): 2669-2684. doi: 10.3934/cpaa.2013.12.2669

Long time dynamics for forced and weakly damped KdV on the torus

1. 

Department of Mathematics, University of Illinois, Urbana, IL 61801, United States

2. 

Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL, 61801

Received  August 2012 Revised  April 2013 Published  May 2013

The forced and weakly damped Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. Starting from $L^2$ and mean-zero initial data we prove that the solution decomposes into two parts; a linear one which decays to zero as time goes to infinity and a nonlinear one which always belongs to a smoother space. As a corollary we prove that all solutions are attracted by a ball in $H^s$, $s\in(0,1)$, whose radius depends only on $s$, the $L^2$ norm of the forcing term and the damping parameter. This gives a new proof for the existence of a smooth global attractor and provides quantitative information on the size of the attractor set in $H^s$. In addition we prove that higher order Sobolev norms are bounded for all positive times.
Citation: M. Burak Erdoğan, Nikolaos Tzirakis. Long time dynamics for forced and weakly damped KdV on the torus. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2669-2684. doi: 10.3934/cpaa.2013.12.2669
References:
[1]

A. V. Babin, A. A. Ilyin and E. S. Titi, On the regularization mechanism for the periodic Korteweg-de Vries equation, Comm. Pure Appl. Math., 64 (2011), 591-648. doi: 10.1002/cpa.20356.

[2]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2003), 31-52. doi: 10.3934/dcds.2004.10.31.

[3]

J. Bourgain, Fourier transform restriction phenomena for cer tain lattice subsets and applications to nonlinear evolution equations. Part II: The KdV equation, GAFA, 3 (1993), 209-262. doi: 10.1007/BF01895688.

[4]

M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation, Physica D, 192 (2004), 265-278 doi: 10.1016/j.physd.2004.01.023.

[5]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp Global Well-Posedness for KdV and Modified KdV on $\mathbb R$ and $\mathbb T$, J. Amer. Math. Soc., 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1.

[6]

M. B. Erdoğan and N. Tzirakis, Global smoothing for the periodic KdV evolution,, to appear in Inter. Math. Res. Not., (). 

[7]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Functional Analysis, 151 (1997), 384-436. doi: 10.1006/jfan.1997.3148.

[8]

J. M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time, J. Diff. Eqs., 74 (1988), 369-390. doi: 10.1016/0022-0396(88)90010-1.

[9]

J. M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations, J. Diff. Eqs., 110 (1994), 356-359. doi: 10.1006/jdeq.1994.1071.

[10]

O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete Contin. Dyn. Syst., 6 (2000), 625-644. doi: 10.1006/jdeq.2000.3763.

[11]

T. Kappeler and P. Topalov, Global wellposedness of KdV in $H^{-1}(T, R)$, Duke Math. J., 135 (2006), 327-360. doi: 10.1215/S0012-7094-06-13524-X.

[12]

C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.

[13]

S. B. Kuksin, "Analysis of Hamiltonian PDEs," Oxford University Press, 2000.

[14]

J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696. doi: 10.1002/cpa.3160380516.

[15]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Phyiscs," Applied Mathematical Sciences, 68, Springer, 1997.

[16]

K. Tsugawa, Existence of the global attractor for weakly damped forced KdV equation on Sobolev spaces of negative index, Commun. Pure Appl. Anal., 3 (2004), 301-318. doi: 10.3934/cpaa.2004.3.301.

[17]

X. Yang, Global attractor for the weakly damped forced KdV equation in Sobolev spaces of low regularity, Nonlinear Differ. Equ. Appl., 18 (2011), 273-285. doi: 10.1007/s00030-010-0095-9.

show all references

References:
[1]

A. V. Babin, A. A. Ilyin and E. S. Titi, On the regularization mechanism for the periodic Korteweg-de Vries equation, Comm. Pure Appl. Math., 64 (2011), 591-648. doi: 10.1002/cpa.20356.

[2]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2003), 31-52. doi: 10.3934/dcds.2004.10.31.

[3]

J. Bourgain, Fourier transform restriction phenomena for cer tain lattice subsets and applications to nonlinear evolution equations. Part II: The KdV equation, GAFA, 3 (1993), 209-262. doi: 10.1007/BF01895688.

[4]

M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation, Physica D, 192 (2004), 265-278 doi: 10.1016/j.physd.2004.01.023.

[5]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp Global Well-Posedness for KdV and Modified KdV on $\mathbb R$ and $\mathbb T$, J. Amer. Math. Soc., 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1.

[6]

M. B. Erdoğan and N. Tzirakis, Global smoothing for the periodic KdV evolution,, to appear in Inter. Math. Res. Not., (). 

[7]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Functional Analysis, 151 (1997), 384-436. doi: 10.1006/jfan.1997.3148.

[8]

J. M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time, J. Diff. Eqs., 74 (1988), 369-390. doi: 10.1016/0022-0396(88)90010-1.

[9]

J. M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations, J. Diff. Eqs., 110 (1994), 356-359. doi: 10.1006/jdeq.1994.1071.

[10]

O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete Contin. Dyn. Syst., 6 (2000), 625-644. doi: 10.1006/jdeq.2000.3763.

[11]

T. Kappeler and P. Topalov, Global wellposedness of KdV in $H^{-1}(T, R)$, Duke Math. J., 135 (2006), 327-360. doi: 10.1215/S0012-7094-06-13524-X.

[12]

C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.

[13]

S. B. Kuksin, "Analysis of Hamiltonian PDEs," Oxford University Press, 2000.

[14]

J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696. doi: 10.1002/cpa.3160380516.

[15]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Phyiscs," Applied Mathematical Sciences, 68, Springer, 1997.

[16]

K. Tsugawa, Existence of the global attractor for weakly damped forced KdV equation on Sobolev spaces of negative index, Commun. Pure Appl. Anal., 3 (2004), 301-318. doi: 10.3934/cpaa.2004.3.301.

[17]

X. Yang, Global attractor for the weakly damped forced KdV equation in Sobolev spaces of low regularity, Nonlinear Differ. Equ. Appl., 18 (2011), 273-285. doi: 10.1007/s00030-010-0095-9.

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